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How many integers less than 1000 can be expressed in the form $$\frac{(x + y + z)^2}{xyz} $$ where $x, y, z$ are integers?

So far, I've attempted substituting certain values of $x, y, z$. For example, setting $y = z = 1$ gives $f(x) = x + 4 + \frac{4}{x}$. I haven't been able to get this over 9 yet.

Ayesha
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    What have you tried? Maybe solving in order to one and use a brute-force algorithm inputting various integer values? Or thinking some more about the necessary relations between those numbers? It seems an interesting problem, one more reason for you to show your efforts. Right now, I won't upvote this. – JMCF125 Jan 11 '14 at 15:44
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    If $y=z=1$, which integers are possible? What have you tried so far? – abiessu Jan 11 '14 at 15:45
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    You have already asked this exact question here. – Old John Jan 11 '14 at 15:47
  • Enumerating over integer values $x \leqslant y \leqslant z$ and picking out integer values of $r = (x+y+z)^2/(x y z)$ possible values of $r$ come out as consecutive integers from 1 to 9 except 7. – Sasha Jan 11 '14 at 16:00
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    @Old John I was told to repost it. – Ayesha Jan 11 '14 at 16:08
  • Then you should rephrase the other question. – JMCF125 Jan 11 '14 at 16:16
  • I did that, it would be appreciated then if a moderator could close this. – Ayesha Jan 11 '14 at 16:19
  • Why, if the this question isn't there anymore? – JMCF125 Jan 11 '14 at 16:29
  • Just for the record, I did not vote to close this as a duplicate, but I did think it is a poor question, since it (originally) showed no effort by the OP, no indication of what has been tried, and no indication of where they were stuck. – Old John Jan 11 '14 at 17:55

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